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On the Spectrum of the Penrose LaplacianMichael Dairyko, Christine Hoffman, Julie Pattyson, Hailee Peck
Cornell University Department of Mathematics Summer Math Institute
Abstract
Since the early 1960’s, aperiodic tilings have been a topic of par-ticular interest to mathematicians. The Penrose tiling is one ex-ample of such a tiling. We present basic subdivision rules forgenerating the Penrose tiling using a set of four Robinson tiles.We use MATLAB to create and store the Penrose tiling as a setof data structures and produce the Laplacian matrix of the tiling.In doing so, we provide insight into the spectrum of the Penrosetiling.
Introduction
Before 1982, crystalline structures were defined as having peri-odic structure. However, in April 1982, Dan Shechtman [7] dis-covered an aluminum-manganese quasicrystalline structure thathad long-range order but did not exhibit the periodic patternsthat characterized crystals. Mathematical interest in periodicitybegan before these discoveries of the physical applications. In1973, Roger Penrose found a set of six “non-square” tiles that tileonly nonperiodically. He later reduced it to a set of four tiles, andthen two tiles.Each of these sets of Penrose tiles generates a nonperiodic tilingwhich, when implemented as a physical structure, is a quasicrys-tal with fivefold symmetry. Raphael Robinson took a set of twoprototiles, a rhombus and diamond, and divided them into a setof four triangles. These tiles, known as Robinson triangles, canalso be used to generate a Penrose tiling. Our work focuses onthe Penrose tiling which is generated in this way [5].
Substitution Method
Prototiles - A set of finite inequivalent tiles (i.e. are not equivalentunder rigid motions, expansions, or contractions) [4].Tiling - An arrangement of tiles, such that their union covers andpacks R2 so that distinct tiles have non-intersecting interiors [4].Aperiodic - A set of prototiles which admits infinitely manytilings of the plane, none of which are periodic [6].Iteration - One application of the substitution method (see below)to a set of tiles.
We use the substitution method to generate the Penrose tiling bystarting with a finite subset of the tiling. We then inflate each tileby 1+
√5
2 and apply subdivision rules to the existing tiles, expand-ing the tiling to cover the plane.
Using MATLAB, we created a function genPenTiling.m thatgenerates finite iterations of the Penrose tiling using this substi-tution method.
Laplacian and Spectrum
Laplacian matrix - In a graph G, let u and v be vertices and dv thedegree of vertex v.
∆(u, v) =
dv if u = v,−1 if u and v are adjacent,0 otherwise. [2]
Spectrum of the Laplacian matrix - Let {λ0 ≤ λ1 ≤ . . . ≤ λn−1}be the set of eigenvalues of ∆(G). This set is denoted as σ(∆(G)),and is called the spectrum of ∆(G) [2].
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
Eigenvalues
Itera
tion
of P
enro
se T
iling
Spectrum of Finite Iterations of Penrose Tiling
Using the MATLAB program tilingLaplacian.m we wereable to find the eigenvalues of the Laplacian matrices for the firstseven iterations of the Penrose tiling. Then using plotMat.m,we plotted the eigenvalues of these iterations.
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
Real Numbers
Prob
abilit
y
Cumulative Distribution Function
The MATLAB program cdf.m outputs the cumulative distribu-tion function of the eigenvalues from each iteration. The graphgives the probability that a given eigenvalue is less than or equalto a certain real-valued number, labeled along the x-axis.
Hausdorff Dimension
Let X ⊂ Rn be nonempty and s be any nonnegative real number.The s-dimensional Hausdorff measure of X, denoted by H s(X),is
limδ→0
H sδ (X)
where δ > 0 and
H sδ (X) := inf
{∞
∑i=1|Ui|s : Ui is a δ-cover of X
}. [3]
The Hausdorff dimension of X, denoted by dimH(X), is
dimH(X) := inf{s : H s(X) = 0} = sup{s : H s(X) = ∞}.
We will have a single point, the Hausdorff dimension, at whichthe measure is not zero or infinity [3].
Values of nδs as Functions of i and ss
i .55 .65 .75 .85 .95 .99 .99999
1 9.56 8.92 8.32 7.76 7.24 7.04 7.002 12.59 10.96 9.54 8.31 7.23 6.84 6.753 17.20 13.97 11.35 9.22 7.48 6.89 6.754 23.28 17.64 13.37 10.13 7.68 6.87 6.685 31.36 22.17 15.68 11.08 7.84 6.82 6.596 41.93 27.66 18.25 12.04 7.94 6.72 6.457 52.08 32.05 19.73 12.14 7.47 6.15 5.86
The MATLAB function ndsTable.m takes a set of eigenvalues ofthe Laplacian and a value s to create a column of values usedto estimate the Hausdorff dimension of the spectrum. Thesecolumns are calculated by nδs, where δ = 1
2i for i ∈ {1, 2, . . . , 7}, nis the number of δ-intervals needed to cover all eigenvalues, ands is an estimate for the Hausdorff dimension (a value between 0and 1).
Results
We are interested in the spectrum of the Penrose tiling as it re-veals certain properties of quasicrystals. By [8], the spectrum ofthe Penrose Laplacian is bounded by eight, which is twice thehighest degree of any vertex. The results on the first seven itera-tions support this.For our specific data, we can estimate the Hausdorff dimensionof the Penrose Laplacian spectrum to be between .85 and .99. Fors less than .85, the table values increase as we move down thecolumns. Likewise, for s greater than 0.99, the table values de-crease down the columns. This implies that the spectrum of thePenrose Laplacian has fractal structure.
Future work
We would like to explore ways to generate more iterations of thePenrose tiling to improve the estimate of the Hausdorff dimen-sion. We hope to use the methods established here to generalizeour results to other aperiodic tilings.
References
[1] A. J. Chorin. Numerical estimates of Hausdorff dimension. J. Comput. Phys.,46(3):390–396, 1982.
[2] F. R. K. Chung. Spectral graph theory, volume 92 of CBMS Regional ConferenceSeries in Mathematics. Published for the Conference Board of the MathematicalSciences, Washington, DC, 1997.
[3] K. Falconer. Hausdorff Measure and Dimension, pages 27–38. John Wiley & Sons,Ltd, 2005.
[4] N. P. Frank. A primer of substitution tilings of the euclidean plane. ExpositionesMathematicae, 26(4):295 – 326, 2008.
[5] M. Gardner. Penrose tiles to trapdoor ciphers. MAA Spectrum. Mathematical As-sociation of America, Washington, DC, 1997. lotsand the return of Dr. Matrix,Revised reprint of the 1989 original.
[6] B. Grunbaum and G. C. Shephard. Tilings and patterns. A Series of Books inthe Mathematical Sciences. W. H. Freeman and Company, New York, 1989. Anintroduction.
[7] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.,53:1951–1953, Nov 1984.
[8] D. Spielman. Spectral graph theory. 2009.
Acknowledgments
The figures related to the substitution method were inspired byand modified from images in [4]. The Hausdorff dimension ta-ble was inspired by [1]. We would like to thank Dr. May Meiand Drew Zemke for their help with this project. We would alsolike to thank the Summer Math Institute and the Mathematics De-partment at Cornell University for the use of their resources. Thiswork was supported by NSF grant DMS-0739338.
Contact Information
• Michael Dairyko, Iowa State [email protected]
• Christine Hoffman, Smith [email protected]
• Julie Pattyson, University of Saint [email protected]
• Hailee Peck, Millikin [email protected]
Advisor: May Mei, Denison University, [email protected]
